Inhalt

The lectures cover the main concepts of nonlinear dynamics, emphasizing characterization of stochastic systems, i.e. nonlinear systems in the presence of random fluctuations. First, the basic ideas of nonlinear dynamics, such as characterization of deterministic nonlinear systems, stability analysis, bifurcations and attractors will be introduced. Further, the main concepts of the theory of stochastic processes will be explained. A special attention will be paid to synchronization and bifurcations in the presence of noise. We will consider the most prominent examples of noise-induced effects, such as stochastic resonance and coherence resonance. These phenomena provide the evidence for constructive counter-intuitive role of random fluctuations in dynamical systems. They will be illustrated by examples from biology and climate. Finally, selected parts of complex networks theory will be presented with a special focus on collective behavior of coupled elements in the presence of random fluctuations. Various applications in different fields of science including physics, chemistry, biology and engineering will be discussed. More specifically, the following topics will be discussed in the lectures: